Water is being pumped into a vertical cylinder of radius 5 meters and height 20 meters at a rate of 3 meters/min. How fast is the water level rising when the cylinder is half full?

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Answer 1 · 2017-02-20T18:14:01

I assume that there is an error and it should be $3$ $3$ $m^{3}$ $m^3$ / $min$

Variables
$V$ = volume of water at time $t$
$h$ = height of water at time $t$
(implicit variable: $t$ = time in minutes)

(Note that the radius of the water is constant $5$ $m$ )

Rates of change

$(dV)/dt = 3$ $m^3$ / $min$
Find $(dh)/dt$ when $h = 10$ $" "$ (When the height of the water is half the height of the cylinder.)

Equation relating the Variables

Volume of a cylinder: $V = pir^2h$

Volume of water:

$V = pi(5)^2h$ or

$V = 25pih$

Equation relating rates of change

(Differentiate both sides of the last equation with respect to $t$ .)

$d/dt(V) = d/dt(25pih)$

$(dV)/dt = 25pi(dh)/dt$

Finish

Substitute what we know, solve for what we want:

$3 = 25pi (dh)/dt$

$(dh)/dt = 3/(25pi)$ $m$ / $min$